Pricing Mortgage Insurance Contracts Using Sdlp-Garch Models

[[abstract]]本研究主要貢獻是在文獻上首度運用兩個恆正之Lévy過程(subordinator)建構出恆正 Lévy過程驅動之Lévy過程(簡稱SDLP)。SDLP包含許多重要Lévy過程，例如Kou (2002) 之雙指數跳躍擴散過程、Madan et al. (1998)之VG過程、Car et al. (2002)之CGMY過程、 Koponen (1995)的truncated Lévy flight過程與Kim et al. (2008)之CTS過程。由於任兩個恆正之Lévy過程可以組成新的SDLP，故運用SDLP概念可建構新的Lévy過程。在第一年計畫中，假設GARCH模型之殘差分配服從SDLP之厚尾高峰分配下，本研究發現SDLP-GARCH模型具有下列特性。第一，條件變異數與正的殘差跳躍以及負的殘差跳躍間具有不對稱的效應。第二，此模型具有條件偏態(conditional skewness)與條件峰態(conditional kurtosis)，此於時間序列之特性相符。第三，SDLP-GARCH模型亦可捕抓報酬率與波動度間之槓桿效應。由於房價波動度對於房屋抵押貸款保險合理保費影響甚鉅，本研究將運用相關房價資料，針對不同的SDLP-GARCH模型進行參數校準與波動度預測。在波動度預測評估上，運用MSE、MAE、HMSE與HMAE等評估準則，針對不同的SDLP-GARCH模型預測之波動度作評量。在第二年計畫中，本研究將運用SDLP-GARCH模型，在文獻上首先假設房價過程為厚尾殘差分配之GARCH模型下，推導出房屋抵押貸款保險之合理保費。運用第一年計畫估計之房價過程相關參數，探討房價過程考量異質變異、條件偏態與條件峰態下，房屋抵押貸款保險合理保費之相關特性。[[abstract]]The main contribution of this project is to construct the subordinator driving Lévy processes (SDLP) by using the difference of two subordinators. The variants of SDLP include many popular Lévy processes, such as double exponential jump diffusion process of Kou (2002), the Variance Gamma (VG) process proposed by Madan et al. (1998), CGMY process proposed by Car et al. (2002), the truncated Lévy flight introduced by Koponen (1995) and the classical tempered stable (CTS) process of Kim et al. (2008). Because any two of the subordinators can create a particular SDLP, the variants of SDLP are up to a maximum number n2 when the total number of subordinators is equal to n. The first-year project is to incorporate the residuals obeying the law of SDLP with GARCH-type models. The SDLP-GARCH models exhibit the following properties. First, the conditional variance is in asymmetric response to the positive jumps and negative jumps of residuals. Second, they can capture the conditional skewness and conditional kurtosis often observed in financial time series. Third, they can capture the leverage effect. Using the housing price data (such as U.S. national average new home price returns for single-family mortgage), I will empirically test the variants of SDLP-GARCH models and use four different accurate statistics―MSE, MAE, HMSE and HMAE―to measure the accuracy of volatility forecast. In the second-year project is to apply the SDLP-GARCH models and to derive the closed-form solutions of fair mortgage insurance premiums, together with conditional Esscher transform for the risk neutral probability measure. Using the parameters of housing pricing data obtained in the first-year project, I will provide the numerical analyses to demonstrate the properties of mortgage insurance contracts and to assess the impacts of leverage effect, conditional skewness and conditional kurtosis on mortgage insurance premiums.[[note]]NSC100-2410-H327-01